POINTWISE MULTIPLIERS FROM THE HARDY SPACE TO
THE BERGMAN SPACE
NATHAN S. FELDMAN
Abstract. For which regions G is the Hardy space H 2 (G) contained in the
Bergman space L2a (G)? This paper relates the above problem to that of finding
the multipliers of H 2 (D) into L2a (D). When G is a simply connected region
this leads to a solution of the above problem in terms of Lipschitz conditions
on the Riemann map of D onto G. For arbitrary regions G, it is shown that
if G is the range of a function whose derivative is a multiplier from H 2 (D) to
L2a (D), then H 2 (G) is contained in L2a (G). Also, if G has a piecewise smooth
boundary, then it is shown that H 2 (G) is contained in L2a (G) if and only if the
angles at all the “corner” points are at least π/2. Examples of multipliers from
H 2 (D) to L2a (D) are given; and in particular, every Bergman inner function is
such a multiplier.
0. Preliminaries
If G is a region in the complex plane C and 1 ≤ p < ∞, then the Bergman space
Lpa (G) is the space of all analytic functions f on G so that |f |p is integrable with
respect to area measure on G. Endowed with the usual Lp norm, Lpa (G) becomes
a Banach space. The Hardy space H p (G) is the space of all analytic functions f
on G so that |f |p has a harmonic majorant. Among all the harmonic majorants of
|f |p there is a smallest one, uf , called the least harmonic majorant. In order to put
a norm on H p (G), first choose a point a ∈ G, then define kf kpH p = uf (a). With
this norm H p (G) becomes a Banach space. For p = 2, both the Bergman space
and Hardy space are separable Hilbert spaces. See [5] and [6] for more information
on Hardy spaces and Bergman spaces, both on the unit disk and in more general
regions.
This paper addresses the problem of characterizing those regions G with the
property that H p (G) is contained in Lpa (G) when 1 ≤ p < ∞. For example, if G
is a bounded by a finite number of smooth curves then H p (G) ⊆ Lpa (G); see [3],
p. 21. More generally, if G is a region so that every positive harmonic function on
G is integrable with respect to area measure, then since functions in H p (G) have
harmonic majorants, H p (G) ⊆ Lpa (G). However, there are simply connected regions
G where H p (G) ⊆ Lpa (G), yet not every positive harmonic function is integrable.
The integrability of positive harmonic functions has been studied by several people;
see [14] and the references there. Also, Axler and Shields in [2] construct a region G
bounded by a rectifiable Jordan curve so that the Dirichlet space is not contained
in the Bergman space on G, thus necessarily H 2 (G) is not contained in L2a (G).
Date: Received September 25, 1996. Appeared: Illinois J. Math. 43 (1999) no. 2, 211–220.
2000 Mathematics Subject Classification. Primary: 30H05; Secondary: 46E15.
Key words and phrases. Hardy Space, Bergman Space, Multiplier.
He was partially supported by the National Science Foundation.
1
2
NATHAN S. FELDMAN
This work relates the containment problem mentioned above to multipliers of
H p (D) into Lpa (D). These multipliers have been characterized by Stegenga [13].
We, however, are interested in a different type of characterization of the multipliers
and it will not depend on his work.
If µ is a positive regular Borel measure on G and X is a Banach space of analytic
functions on G, then we say that µ is a p-Carleson measure for X, if X ⊆ Lp (µ). The
usual definition of a Carleson measure, in our terminology, is simply a 2-Carleson
measure for H 2 (D). It is well known that if µ is a measure on the open unit disk D,
then µ is a p-Carleson measure for H p (D) if and only if there exists a constant C
such that µ(Sh (θ)) ≤ Ch for every Carleson square Sh (θ) = {z ∈ D : 1−|z| ≤ h and
θ − h ≤ arg(z) ≤ θ + h}. See [7], p. 156 or [9], p. 33 for a proof. In particular since
the condition above on Carleson squares is independent of p, we see that a measure
µ is a p-Carleson measure for H p (D) if and only if µ is a 2-Carleson measure for
H 2 (D).
1. Basic properties of multipliers
In this section a relation is given between the containment of the Hardy space
inside the Bergman space and multiplication operators that map H p (D) into Lpa (D).
Also certain growth conditions are given for functions that multiply H p (D) into
Lpa (D).
Proposition 1.1. Suppose G is a simply connected region and τ : D → G is a
Riemann map.
Then for 1 ≤ p < ∞ the following are equivalent:
(a) H p (G) ⊆ Lpa (G);
(b) (τ 0 )2/p H p (D) ⊆ Lpa (D); that is (τ 0 )2/p f ∈ Lpa (D)
for each f ∈ H p (D);
(c) µ = |τ 0 |2 dA is a p-Carleson measure for H p (D).
Proof. Let h ∈ H p (G). Since τ is univalent, by the usual change of variables we
have
Z
Z
2
p
p
|h| dA =
|h ◦ τ | |τ 0 | dA.
(∗)
G
D
Now as h varies over all of H p (G), h ◦ τ varies over all of H p (D), since the Hardy
spaces are conformally invariant. So, if (a) holds then each f in H p (D) has the
form f = h ◦ τ for some h in H p (G), thus (τ 0 )2/p f = (h ◦ τ )(τ 0 )2/p and the Lpa (D)
norm of this last expression is exactly the right hand side of (∗). But by (a), the
left hand side of (∗) is finite, so (a) implies (b).
if (b) holds, then for each f in H p (D), (τ 0 )2/p f ∈ Lpa (D). Thus
R Now
p 0 2
|f | |τ | dA < ∞ and hence H p (D) ⊆ Lp (µ), where µ = |τ 0 |2 dA. So µ is a
D
p-Carleson measure for H p (D) and (c) follows. Finally, if (c) holds, then the right
hand side of (∗) is finite for each h in H p (G). Hence the left hand side is also and
so (a) follows.
Corollary 1.2. If 1 ≤ p < ∞ and G is a simply connected region, then H p (G) ⊆
Lpa (G) if and only if H 2 (G) ⊆ L2a (G).
This can be seen by checking condition (b) above or by observing as mentioned
before that a measure µ is a p-Carleson measure for H p (D) if and only if µ is a
Carleson measure for H 2 (D).
POINTWISE MULTIPLIERS FROM THE HARDY SPACE TO THE BERGMAN SPACE
3
In view of Corollary 1.2 we will mainly restrict our attention to p = 2, occasionally considering or commenting on other values of p ≥ 1.
Proposition 1.1 makes precise the relation between the containment of H p (G) in
p
La (G) and multipliers on the unit disk. Notice that when p = 2 and τ : D → G
is a Riemann map then H 2 (G) ⊆ L2a (G) if and only if τ 0 is a multiplier of H 2 (D)
into L2a (D). Thus we shall try to understand which functions multiply H 2 (D) into
L2a (D). In particular, motivated by the univalent case, we want to understand
which functions φ have the property that φ0 is a multiplier of H 2 (D) into L2a (D).
Unless otherwise stated, whenever we say multiplier we mean an analytic function
on the unit disk that multiplies H 2 (D) into L2a (D). Let M (H 2 , L2a ) denote the set
of multipliers. Each multiplier f induces a multiplication operator, denoted by Mf .
Theorem 1.3. (a) If f is a multiplier of H 2 (D) into L2a (D), then f ∈ L2a (D) and
there is a constant C such that (1 − |z|2 )|f (z)|2 ≤ C for all z in D.
(b) If f is any analytic function on D, then fR is a multiplier of H 2 (D) into L2a (D)
if and only if there is a constant K so that Sh |f |2 dA ≤ Kh for each Carleson
square Sh of size h.
Proof. (a) If f ∈ M (H 2 , L2a ), then, since the constants are in H 2 (D), f must be
2 1/2
2
)
)
and Bw = (1−|w|
in L2a (D). Let Kw = (1−|w|
(1−w̄z)
(1−w̄z)2 be the normalized reproducing
kernels in H 2 (D) and L2a (D), respectively. Since f Kw is in L2a (D) and (1−1w̄z)2 is
the reproducing kernel in L2a (D), for each w in D we have
2
2
|hf Kw , Bw i| = (1 − |w| ) |(f Kw )(w)| = (1 − |w| )1/2 |f (w)| .
It follows that for all w in D,
2
(∗∗)(1 − |w| )1/2 |f (w)| = |hf Kw , Bw i| = |hMf Kw , Bw i| ≤ kMf k
Hence (a) holds.
suppose f is analytic on D, then f ∈ M (H 2 , L2a ) if and only if
R Now
2
|h| |f |2 dA < ∞ for each h ∈ H 2 (D). But this is equivalent to µ = |f |2 dA
D
being a Carleson measure and condition (b) is exactly the condition mentioned in
the previous section that characterizes Carleson measures.
Recall that the Dirichlet space D(G) consists of all analytic functions on G whose
derivative is in L2a (G). Also a function f defined on a region G is Lipschitz of order
α if there is a constant C so that |f (z) − f (w)| ≤ C|z − w|α for all z, w ∈ G.
Corollary 1.4. If ϕ is an analytic function on D and ϕ0 ∈ M (H 2 , L2a ), then ϕ is
in the Dirichlet space and is Lipschitz of order 1/2.
Proof. Clearly ϕ0 ∈ L2a (D) as ϕ0 is a multiplier, so ϕ is in the Dirichlet space. The
fact that ϕ is Lipschitz of order 1/2 follows from [7], p. 74 since Theorem 1.3 (a)
C
implies that there is a constant C such that |ϕ0 (z)| ≤ (1−|z|
2 )1/2 .
Corollary 1.5. Suppose G is a simply connected region and τ : D → G is a Riemann map. If H 2 (G) ⊆ L2a (G) , then τ is Lipschitz of order 1/2. In particular τ
is continuous on the closed unit disk and G is bounded.
The previous corollary gives many examples of simply connected regions where
H 2 (G) is not contained in L2a (G). For example, such a containment fails whenever ∂G is not locally connected, because then the Riemann map will not extend
continuously to ∂D.
4
NATHAN S. FELDMAN
Corollary 1.5 also raises an interesting question about the boundedness of G,
when G is not simply connected. Namely, if G is any region such that H 2 (G) ⊆
L2a (G), does G have to be bounded? Clearly since H 2 (G) contains the constants,
G must have finite area. As noted above, if G is simply connected, then G must
be bounded, but for arbitrary regions of finite area it is not clear that boundedness
holds.
By a compact multiplier we mean a function f whose multiplication operator
Mf is compact.
Theorem 1.6. (a) If f induces a compact multiplier of H 2 (D) into L2a (D), then
(1 − |z|2 )|f (z)|2 → 0 as |z| → 1.
(b) If f is an analytic function
R on D, then f induces a compact multiplier of
H 2 (D) into L2a (D) if and only if Sh |f |2 dA = o(h) as h → 0.
Proof. (a) Consider the normalized reproducing kernels, Kw and Bw as in Theorem 1.3. Since Mf is compact and Kw converges weakly to zero as |w| → 1,
we have that f Kw converges to zero in norm in L2a (D). Thus since Bw has norm
one, |hMf Kw , Bw i| → 0 as |w| → 1, thus equation (∗∗) above gives the desired
conclusion.
To see that (b) holds, notice that Mf is compact if and only if the inclusion of
H 2 (D) into L2 (|f |2 dA) is compact, and this means, by definition, that |f |2 dA is a
“vanishing Carleson measure”. Further, these measures are characterized exactly
by the condition stated in (b); see [15], p. 172.
Notice that the estimate in (a) of Theorems 1.3 and 1.6 can also be proved by
using the Carleson measure estimate, part (b) from Theorems 1.3 and 1.6, together
with the subharmonicity of |f |2 . This technique works for other values of p where
reproducing kernel arguments are not as readily available.
We are interested in finding to what extent the necessary conditions of Corollary 1.4 are sufficient to guarantee that ϕ0 is a multiplier. We shall see that a
stronger condition holds on the valence of ϕ. Although, if ϕ is univalent, the conditions of Corollary 1.4 are both necessary and sufficient.
2. The valence function
In Corollary 1.4 it was shown that if ϕ is analytic on D and ϕ0 is a multiplier,
then ϕ is in the Dirichlet space and is Lipschitz of order 1/2.
In this section it is shown that the converse holds for a large class of functions,
including the univalent ones.
In order to do this, we need a change of variables formula for non-univalent
functions. So suppose ϕ : G → C is an analytic function on an open set G. Define
its valence function or counting function nϕ (w) as the number of points, counting
multiplicity, in the pre-image, ϕ−1 (w). So nϕ (w) is defined on all of C, but is
zero off the range of ϕ. Also, notice that ϕ is univalent precisely when nϕ (w) ≤ 1
everywhere on C and that nϕ (w) may equal infinity. The following is a change
of variables formula for non-univalent functions. It is well known in geometric
measure theory and is useful in the study of analytic functions. We include a proof
for completeness.
POINTWISE MULTIPLIERS FROM THE HARDY SPACE TO THE BERGMAN SPACE
5
Theorem 2.1. Suppose ϕ : G → Ω is a non-constant analytic function with valence function nϕ (w). If f : Ω → [0, ∞) is any Borel function, then
Z
Z
2
f (ϕ(z)) |ϕ0 (z)| dA(z) =
f (w)nϕ (w)dA(w).
Ω
G
0
Proof. If Z = {z ∈ G : ϕ (z) = 0}, then Z is a discrete set in G and hence has
area zero. So ϕ is univalent on some small disk about each point of G − Z. Using
Vitali’s covering lemma we can find a sequence of disjoint
S∞ disks Bn inside G so
that ϕ|Bn is univalent for each n and the area of G − Pn=1 Bn is zero. If χE is
∞
the characteristic function of the set E, then nϕ (w) = n=1 χϕ(Bn ) a.e. (area).
This is because the Bn ’s cover almost all of G and since analytic functions always
map sets of area zero onto sets of area zero, their images ϕ(Bn ) cover almost
all of the range of ϕ. Also, this expression for nϕ shows that it is a measurable
the usual change of variables
function. So for
R
R each n, since ϕ|Bn is univalent,
formula gives Bn f (ϕ(z))|ϕ0 (z)|2 dA(z) = ϕ(Bn ) f (w)dA(w). Since the disks are
pairwise disjoint, we get
Z
∞ Z
∞ Z
X
X
2
0
f (ϕ(z)) |ϕ (z)| dA(z) =
f (w)dA(w) =
f (w)χϕ(Bn ) dA(w)
G
n=1
Z
=
ϕ(Bn )
f (w)
Ω
∞
X
n=1
n=1
Z
Ω
χϕ(Bn ) dA(w) =
f (w)nϕ (w)dA(w).
Ω
Notice that we are allowed to change the integral and the sum because everything
is positive.
Corollary 2.2. If ϕ : RG → Ω is a non-constant
analytic function and nϕ is its
R
valence function, then G |ϕ0 (z)|2 dA(z) = Ω nϕ (w)dA(w).
Notice this says that ϕ is in the Dirichlet space if and only if its valence function
is an L1 function. Thus assuming a function is in the Dirichlet space is simply
imposing a restriction on the growth of its valence function.
Now, consider an analytic function ϕ on D that is Lipschitz of order 1/2 and in the
Dirichlet space. Is such a function a multiplier? If we impose a stronger condition
on the valence of the function ϕ, then the Lipschitz condition will guarantee that
ϕ is a multiplier. We will consider the case when nϕ is essentially bounded, that
is, there is a constant C so that nϕ (w) ≤ C for all w except a set of area zero.
Theorem 2.3. If ϕ is analytic on D and nϕ is essentially bounded, then ϕ0 is a
multiplier if and only if ϕ is Lipschitz of order 1/2.
Proof. In view of Corollary 1.4 and Theorem 1.3 (b) it suffices to show that if ϕ
is Lipschitz of order 1/2 then µ = |ϕ0 |2 dA is a Carleson measure. So, if Sh is a
Carleson square of size h, then by Corollary 3.2 we have
Z
Z
2
|ϕ0 | dA =
nϕ (w)dA
µ(Sh ) =
Sh
≤
ϕ(Sh )
knϕ k∞ Area{ϕ(Sh )} ≤ π knϕ k∞ diam{ϕ(Sh )}2 ≤ Ch
where the constant C depends only on the norm of nϕ and the Lipschitz constant
of ϕ. Thus µ is a Carleson measure on D.
6
NATHAN S. FELDMAN
Notice that the assumption that nϕ is essentially bounded is stronger than assuming that ϕ is only in the Dirichlet space, that is nϕ ∈ L1 .
But there is still an ample supply of such functions. First, all the univalent
functions are in this class and also for any bounded region G, there is an analytic
function ϕ on D so that ϕ(D) = G and nϕ is essentially bounded. So such functions
come in all shapes and sizes.
Corollary 2.4. If ϕ is a univalent function on D, then ϕ0 is a multiplier of H 2 (D)
into L2a (D) if and only if ϕ is Lipschitz of order 1/2.
Corollary 2.5. If G is a simply connected region and τ : D → G is a Riemann
map, then H 2 (G) ⊆ L2a (G) if and only if τ is Lipschitz of order 1/2.
Smith and Stegenga in [12] have given a geometric characterization, in terms
of the hyperbolic metric, of simply connected regions for which the Riemann map
satisfies a Lipschitz condition.
Now suppose that G is bounded by a finite number of piecewise C 1 -smooth
disjoint Jordan curves, each having one-sided derivatives at the “corner points”.
Consider the angles θj , where 0 ≤ θj ≤ 2π, formed by the tangent lines at the
corner points. So, if θj = 0 at a corner point w, then there is an outward cusp at
w and if θj = 2π there is an inward cusp at w. However, if θj = π, then the curve
is actually smooth at w.
Corollary 2.6. If G is as above, then H 2 (G) ⊆ L2a (G) if and only if θj ≥ π/2 for
all j.
Proof. First suppose that G is simply connected and bounded by a piecewise smooth
Jordan curve having corners at the points {wj } forming angles {θj }. Let τ : D → G
be a Riemann map and zj ∈ ∂D satisfy τ (zj ) = wj . If the angle θj = αj π where 0 ≤
αj ≤ 2, then (z −zj )1−αj τ 0 (z) has a non-zero finite limit at zj ; see Pommerenke[11],
p. 52. However as previously mentioned, τ is Lipschitz of order 1/2 if and only if
C
|τ 0 (z)| ≤ (1−|z|
2 )1/2 holds for some constant C; see [7], p. 74. Thus τ is Lipschitz of
order 1/2 if and only if αj ≥ 1/2 for all j. This, together with Corollary 2.5, gives
the desired conclusion in this case. The general case when G is finitely connected
may be reduced to the simply connected case because H 2 (G) may be decomposed
as a direct sum of Hardy spaces over simply connected regions in a canonical way;
see Conway [4].
Example If G is a triangle, then H 2 (G) 6⊂ L2a (G); however if G is a rectangle,
then H 2 (G) ⊆ L2a (G). If G has an outward cusp (θ = 0), then H 2 (G) 6⊂ L2a (G).
It is interesting that it is rather difficult to construct a region G bounded by a
rectifiable Jordan curve such that the Dirichlet space on G is not contained in the
Bergman space on G, see Axler and Shields [2].
Now it is shown that there is a slightly stronger necessary condition on an analytic function ϕ in order to have ϕ0 a multiplier. Corollary 1.4 shows that if ϕ0
is a multiplier, then ϕ is in the Dirichlet space; that is, nϕ ∈ L1 . Thus, if we let
µ = nϕ (w)dA(w), then Corollary 1.4 says that µ is a finite measure whenever ϕ0
is a multiplier. It is now shown that whenever ϕ0 is a multiplier µ is a Carleson
measure for H 2 (G), where G = ϕ(D).
POINTWISE MULTIPLIERS FROM THE HARDY SPACE TO THE BERGMAN SPACE
7
Theorem 2.7. Suppose ϕ is analytic on D and G = ϕ(D). If ϕ0 is a multiplier of
H 2 (D) into L2a (D), then the measure µ = nϕ dA is a Carleson measure for H 2 (G).
That is, H 2 (G) ⊆ L2 (µ).
Proof. Let h ∈ H 2 (G). Since ϕ : D → G is analytic, we have h(ϕ(z)) ∈ H 2 (D) and,
because ϕ0 is a multiplier, h(ϕ(z))ϕ0 (z) ∈ L2a (D). So by Theorem 2.1 we have
Z
Z
2
2
2
|h(w)| nϕ (w)dA(w) =
|h(ϕ(z))| |ϕ0 (z)| dA(z) < ∞.
G
2
D
Thus, h ∈ L (µ) and so µ is a Carleson measure for H 2 (G).
This theorem has a very nice corollary.
Corollary 2.8. Suppose ϕ is an analytic function on D and G = ϕ(D). If ϕ0 is a
multiplier, then H 2 (G) ⊆ L2a (G).
Proof. Since nϕ ≥ 1 on G the theorem gives H 2 (G) ⊆ L2a (G, nϕ dA) ⊆ L2a (G).
Theorem 2.3 and Corollary 2.4 give some examples of multipliers to which Corollary 2.8 may be applied. Also, it will be shown in Theorem 3.3 that if f is any
function in H 2 (D) or L4a (D), then f is a multiplier. Hence if ϕ is a primitive of f ,
then Corollary 2.8 applies to give H 2 (G) ⊆ L2a (G) for G = ϕ(D). See Corollary 3.4.
3. Examples
In this section some examples of multipliers are given. The main tool used to
show functions are multipliers is the Carleson measure condition of Theorem 1.3.
Since H 2 (D) ⊆ L2a (D) it is clear that every bounded analytic function on D is a
multiplier. We next show that much more is true. Recall that Theorem 1.3 shows
C
that if f ∈ M (H 2 , L2a ), then |f (z)| ≤ (1−|z|
2 )1/2 . In Theorem 3.1 below we show
that a slight improvement on this condition implies that f is a compact multiplier.
Theorem 3.1. If f is analytic on D and |f (z)| ≤ ρ(|z|), where ρ ∈ L2 (0, 1), then
f induces a compact multiplier from H 2 (D) to L2a (D).
Proof. Let {gn } ⊆ H 2 (D) and suppose gn → 0 weakly. So {gn } is norm bounded
in H 2 (D) and gn → 0 uniformly on compact subsets of D. We must show that
kf gn kL2 → 0 as n → ∞. So
Z 1 Z 2π
Z 1 Z 2π
¯
¯
¯
¯
2
iθ
iθ ¯2
¯
¯gn (reiθ )¯2 dθρ(r)2 r dr
f (re )gn (re ) dθr dr ≤
kf gn kL2 =
0
Z
0
0
0
1
M2 (gn , r)2 ρ(r)2 r dr,
=
0
R 2π
where M2 (gn , r)2 = 0 |gn (reiθ )|2 dθ. But, M2 (gn , r)2 ρ(r)2 r → 0 as n → ∞ for
each r, because gn → 0 uniformly on compact sets in D. Also, there is a constant
C so that M2 (gn , r)2 ≤ C for all n and r, because the set {gn } is norm bounded in
H 2 (D). So the integrand of the last integral above is dominated by an integrable
function, namely Cρ2 . Thus the Lebesgue Dominated Convergence Theorem says
that the last integral goes to zero as n tends to infinity.
Corollary 3.2. If f is analytic on D and |f (z)| ≤ (1−|z|C2 )1/2−ε for some ε, C > 0
and all z in D, then f induces a compact multiplier.
8
NATHAN S. FELDMAN
The previous results give a large number of examples of compact multipliers
and Theorem 3.3 below even gives more. However, not all multipliers are compact
1
because f (z) = (1−z)
1/2 is a multiplier that is not compact, see Theorem 1.6.
Theorem 3.3. (a) H 2 (D) ⊆ L4a (D).
(b) Each function in L4a (D) induces a compact multiplier.
Proof. (a) It is a classic result due to Hardy and Littlewood that H p (D) ⊆ L2p
a (D);
see Duren [7], p. 87.
(b) If f ∈ L4a (D) and g ∈ H 2 (D) , then kf gkL2 ≤ kf kL4 kgkL4 ≤ ckf kL4 kgkH 2 .
Thus we see that f ∈ M (H 2 , L2a ) and kMf k ≤ Ckf kL4 for an absolute constant C.
It follows from Theorem 3.1 that every polynomial induces a compact multiplication
operator. Since the polynomials are dense in L4a (D), we may choose pn → f in
L4a (D). Thus from the above estimate we see that Mpn → Mf in operator norm,
hence Mf is compact.
The following result follows from Theorem 3.4 and Corollary 2.8.
Corollary 3.4. If ϕ is an analytic map on the disk D, G = ϕ(D) and ϕ0 ∈ L4a (D),
then H 2 (G) ⊆ L2a (G).
R
A function ϕ in L2a (D) is a Bergman inner function if D u(z)|ϕ(z)|2 dA = u(0)
for every bounded harmonic function u on D; see [1] or [8]. It is known that if
M is an invariant subspace of the Bergman shift and ϕ ∈ M ∩ (zM)⊥ has norm
one, then ϕ is a Bergman inner function. Next we show that every Bergman inner
function is a multiplier. This also appears in Hedenmalm’s [10] work, although this
proof is easier and more direct.
Theorem 3.5. Every Bergman inner function induces a contractive multiplier.
R
Proof. Let ϕ be a Bergman inner function. Thus D u(z)|ϕ(z)|2 dA = u(0) for
every bounded harmonic function u on D. If u is a positive harmonic function on
D and ur (z) R= u(rz), then ur → u pointwise as r → 1, so Fatou’s Lemma easily
implies that D u(z)|ϕ(z)|2 dA ≤ u(0) for all positive harmonic functions on D. If
f ∈ H 2 (D) and uf is the least harmonic majorant for |f |2 , then we have
Z
Z
2
2
2
|f (z)ϕ(z)| dA ≤
uf (z) |ϕ(z)| dA ≤ uf (0) = kf kh2 .
D
D
Thus, ϕ is a multiplier and kMϕ k ≤ 1.
We close with a few natural questions that remain open.
Question 1 If G is a region such that H 2 (G) ⊆ L2a (G), then must G be bounded?
Question 2 If G is a region such that H 2 (G) ⊆ L2a (G), then must there exist
an analytic function ϕ on D with ϕ0 a multiplier and ϕ(D) = G? That is, is the
converse of Corollary 2.8 true?
Question 3 If ϕ is an analytic function on D that is Lipschitz of order 1/2 and
nϕ dA is a Carleson measure for H 2 (G), G = ϕ(D), then must ϕ0 be a multiplier?
Notice that Questions 1 and 2 both have an affirmative answer when G is simply
connected.
The author would like to thank the referee for many helpful suggestions in writing
this paper.
POINTWISE MULTIPLIERS FROM THE HARDY SPACE TO THE BERGMAN SPACE
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Michigan State University,
East Lansing, MI 48824
E-mail address: [email protected]
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